Problem: All of the 3rd grade teachers and students from Loyola went on a field trip to an art museum. Tickets were $$6.00$ each for teachers and $$4.50$ each for students, and the group paid $$55.50$ in total. A few weeks later, the same group visited a natural history museum where the tickets cost $$18.00$ each for teachers and $$9.50$ each for students, and the group paid $$138.50$ in total. Find the number of teachers and students on the field trips.
Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${6x+4.5y = 55.5}$ ${18x+9.5y = 138.5}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-3$ ${-18x-13.5y = -166.5}$ ${18x+9.5y = 138.5}$ Add the top and bottom equations together. $ -4y = -28 $ $ y = \dfrac{-28}{-4}$ ${y = 7}$ Now that you know ${y = 7}$ , plug it back into $ {6x+4.5y = 55.5}$ to find $x$ ${6x + 4.5}{(7)}{= 55.5}$ $6x+31.5 = 55.5$ $6x = 24$ $x = \dfrac{24}{6}$ ${x = 4}$ You can also plug ${y = 7}$ into $ {18x+9.5y = 138.5}$ and get the same answer for $x$ ${18x + 9.5}{(7)}{= 138.5}$ ${x = 4}$ There were $4$ teachers and $7$ students on the field trips.